While solving a mathematical problem, it might be that we don’t have enough of an existing understanding to deal with it. Sometime we can’t believe a given situation (a described agreement) is possible simply because of our lack of experience in that particular area of maths. In those cases we have to push ourselves forward to expand our learning. It’s important not to give up; the only way you learn is by comparing what you don’t know with what you do until you also know that. You have to work through all the examples, despite the fact you are trying to prove something that you think is actually incorrect, until you reach a definite conclusion one way or another. Unless you have rigorously worked through every possible consideration you cannot say for sure that it wasn’t possible. Sochiro Honda, of Honda Motors, said ”Success is 99% failure” – so giving up is not an option!

Problems:

There are four numbers written in a row. The first number is 100. It is known that the first number if divided by the second number is a prime number, the second number if divided by the third number is a prime number, and the third number if divided by the fourth number is also a prime number. Can all these prime numbers be distinct?

A maths teacher draws a number of circles on a piece of paper. When she shows this piece of paper to the young mathematician, he claims he can see only five circles. The maths teacher agrees. But when she shows the same piece of paper to another young mathematician, he says that there are exactly eight circles. The teacher confirms that this answer is also correct. How is that possible and how many circles did she originally draw on that piece of paper?

A group of three smugglers is offered to smuggle a chest full of treasures across the dangerous river. The boat they possess is old and frail. It can carry three smugglers without the chest, or it can carry the chest and only two smugglers. The price for this job is extremely high, and the gang is more than interested in completing the job. Think of a strategy the smugglers should follow to successfully transit the chest and themselves to the other shore.

My mum once told me the following story: she was walking home late at night after sitting in the pub with her friends. She was then surrounded by a group of unfriendly looking people. They demanded: “money or your life?!” She was forced to give them her purse. She valued her life more, since she was pregnant with me at that time. According to her story she gave them two purses and two coins. Moreover, she claimed that one purse contained twice as many coins as the other purse. Immediately, I thought that the mum must have made a mistake or could not recall the details because of the shock and the amount of time that passed after that moment. But then I figured out how this could be possible. Can you?